Question

Question is in the attachment.
please answer.
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Answer 1

Information provided with us :

• $\sf{A.P. \: is \: \: 20, \: 19\dfrac{1}{3} \: , \: 18\dfrac{2}{3} ...}$
• Sum of that A.P. is 300

What we have to calculate :

• Number of terms of that given A.P. ?

Using Formulas :

Sum of n terms of an A.P. is calculated by,

• $\red{\boxed{\bf{S_{n} = \dfrac{n}{2} \: [2a + (n - 1)d ]}}}$

Here,

• n is number of terms
• d is common difference
• a is first term of the A.P.

Performing Calculations :

Finding out common difference (d) :-

$\implies \sf{d \: = \: 19 \dfrac{1}{3} - 20 }$

$\implies \: \sf{d \: = \: 19 \dfrac{1}{3} - \dfrac{20}{1} }$

$\implies \: \sf{d \: = \: \dfrac{19 \times 3 + 1}{3} - \dfrac{20}{1} }$

$\implies \: \sf{d \: = \: \dfrac{57+ 1}{3} - \dfrac{20}{1} }$

$\implies \: \sf{d \: = \: \dfrac{58}{3} - \dfrac{20}{1} }$

$\implies \: \sf{d \: = \: \dfrac{58}{3} - \dfrac{20 \times 3}{1 \times 3} }$

$\implies \: \sf{d \: = \: \dfrac{58 - 60}{3} }$

$\implies \: \boxed{\sf{d \: = \: \dfrac{ - 2}{3} }}$

Putting the values :

$: \longmapsto \: 300 = \dfrac{ \sf{n}}{2} [(2)(20) + ( \sf{n} - 1) - \dfrac{1}{2}$

$: \longmapsto \: 300 = \dfrac{ \sf{n}}{2} [(2 \times 20) + ( \sf{n} - 1) - \dfrac{1}{2}$

$: \longmapsto \: 300 = \dfrac{ \sf{n}}{2} [40 + ( \sf{n} - 1) - \dfrac{1}{2}$

$: \longmapsto \: 300 = \dfrac{ \sf{n}}{2} \bigg[40 - \dfrac{2}{3}n \: + \dfrac{2}{3} \bigg ]$

$: \longmapsto \: 300 \times 2 = n \bigg[40 - \dfrac{2}{3}n \: + \dfrac{2}{3} \bigg ]$

$: \longmapsto \: 600= n \bigg[40 - \dfrac{2}{3}n \: + \dfrac{2}{3} \bigg ]$

$: \longmapsto \: 600= n \bigg[ \dfrac{40}{1} - \dfrac{2}{3}n \: + \dfrac{2}{3} \bigg ]$

$: \longmapsto \: 600= n \bigg[ \dfrac{40 \times 3}{1 \times 3} - \dfrac{2}{3}n \: + \dfrac{2}{3} \bigg ]$

$: \longmapsto \: 600= n \bigg[ \dfrac{120 - 2n + 2}{3} \bigg ]$

$: \longmapsto \: 600= n \times \bigg[ \dfrac{120 - 2n + 2}{3} \bigg ]$

$: \longmapsto \: \sf{600 \times 3= 120n- 2n {}^{2} + 2n}$

$: \longmapsto \: \sf{1800= 120n- 2n {}^{2} + 2n}$

$: \longmapsto \: \sf{1800= 122n- 2n {}^{2} }$

$: \longmapsto \: \sf{ 2n {}^{2} - 122n + 1800 = 0}$

$: \longmapsto \: \sf{ n {}^{2} - 61n + 900 = 0}$

$: \longmapsto \: \sf{(n - 25)(n - 36) = 0}$

On comparing them we gets 25 and 36.

$\underline{ \bf{Henceforth, \: no. \: of \: terms \: are \: 25 \: or \: 36}}$

Answer 2

Given:-

Ap is in the attachment

To Find:-

No. of terms

Solution:-

Refer the attachment